It was a pleasure Catullus. This is really an interesting topic.
If you want you can try to share your own thoughts on it, it doesn't matter if you have not results or proofs. It is enough that you applied to it with passion, maybe you have questions or you have a personal way to approach the problem and you look for tips and observations.
Anyways... if you really want something cool to think about, I suggest you to go deeper instead into the functions \(F_a(n)=[a]_n=a \,{\rm mod}\, n\) for varying \(n\).
Only understanding those is very challenging... maybe iterating them would be an even greater challenge.
Just some hints: every function \([a]_n\in \mathbb Z/n\mathbb Z\) belongs to a different ring... so how we obtain \(F_a\)? Just imagine \(F_a\) to take inputs in \(\mathbb Z\) and give output in the union of all the "modular rings" \(\bigcup _{n\in\mathbb N}\mathbb Z/n\mathbb Z\). Why it is interesting? Because the zeros of \(F_a\) are where \(a\) has it's prime decomposition, i.e. \(F_a\) detects \(a\) prime divisors.
If instead you wanto to continue investigating functions \(\pi_n(a)=a\, {\rm mod} \,n\) then you should begin to study what it means to iterate idempotent functions. A function is idempotent iff \(f(f(x))=f(x)\)... this means that it has all fixed points or pre-periodic point.
If you want you can try to share your own thoughts on it, it doesn't matter if you have not results or proofs. It is enough that you applied to it with passion, maybe you have questions or you have a personal way to approach the problem and you look for tips and observations.
Anyways... if you really want something cool to think about, I suggest you to go deeper instead into the functions \(F_a(n)=[a]_n=a \,{\rm mod}\, n\) for varying \(n\).
Only understanding those is very challenging... maybe iterating them would be an even greater challenge.
Just some hints: every function \([a]_n\in \mathbb Z/n\mathbb Z\) belongs to a different ring... so how we obtain \(F_a\)? Just imagine \(F_a\) to take inputs in \(\mathbb Z\) and give output in the union of all the "modular rings" \(\bigcup _{n\in\mathbb N}\mathbb Z/n\mathbb Z\). Why it is interesting? Because the zeros of \(F_a\) are where \(a\) has it's prime decomposition, i.e. \(F_a\) detects \(a\) prime divisors.
If instead you wanto to continue investigating functions \(\pi_n(a)=a\, {\rm mod} \,n\) then you should begin to study what it means to iterate idempotent functions. A function is idempotent iff \(f(f(x))=f(x)\)... this means that it has all fixed points or pre-periodic point.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)